Um breve estudo sobre a curvatura média e o teorema de Aleksandrov

Abstract

When looking for the eigenvalues of the differential of the normal Gauss map dN, naturally arise in its characteristic polynomial two functions that are invariant by change of base of this operator: the determinant of the matrix of the normal Gauss map, called Gaussian curvature and the trace of this application. As this linear map is self-adjoint,there is an orthonormal basis in which its matrix is written diagonally in terms of the principal curvatures, and its determinant and trace are given by det(dN) = (−k1)(−k2) and its dash by tr(dN) = −(k1+k2). The negative half of the H = k1 + k2/2 is the so-called mean curvature, which was introduced by French mathematician Sophie Germain when studying a problem related to membrane vibrations. At this time, a problem proposed by Lagrange, which later received the name of Plateau’s problem, a Belgian physicist who carried out several experiments and in-depth studies on soap films around 1850, was, roughly speaking, to determine a surface that has the smallest area among those which have the edge given by a prescribed Jordan curve. It can be shown that such a surface has zero mean curvature at its regular points. Such surfaces are called minimal and are named after Lagrange. In this work we will make a brief study on mean curvature and minimal surfaces,demonstrating some results and presenting some examples of such surfaces. Furthermore, we will demonstrate Aleksandrov’s theorem which under certain assumptions says that the only compact surface with constant mean curvature in R3 is the sphere. For this, we will demonstrate this result with a different “machinery” the one used by Aleksandrov. We will follow R. Reilly’s approach in his article “Mean Curvature, the Laplacian and Soap Bubbles” which makes use of more basic knowledge of differential and integral calculus and surface theory for its demonstration.